Analytical Linearization of a State-space Free Vortex Wake Model

This paper presents the development and application of analytical linearization of a State-Space Free Vortex Wake Model. Previous work developed a state-space free wake model that could be numerically linearized via finite differences into a Linear Time Periodic (LTP) system, but the numerical linearization process was computationally expensive. An improved method is developed that uses exact analytical linearization of the Biot-Savart Law. The analytical method is found to speed up linearization computations by O(N), where N is the number of free wake nodes. A simple decoupled wake model is used to develop and test the method, where the wake system's inputs are prescribed blade bound circulations. The state space matrices computed by the analytical linearization method are verified to match those of the numerical linearization method exactly as perturbation sizes approach zero. The analytically linearized LTP model was converted into a Linear Time Invariant (LTI) model using Harmonic Decomposition. The on-blade induced velocities predicted by the LTI wake model are found to match the non-linear wake model in both frequency response and for small amplitude step inputs to the blade circulation. The linearized wake model is then coupled with a non-linear blade element simulation of a utility helicopter rotor (using the GenHel rotor model). Step responses to rotor collective and cyclic pitch inputs are compared between the rotor model coupled with the linearized wake and with the non-linear wake model. The rotor forces, hub moments, and induced velocities of the two models are found to have good agreement.